To Barack Obama and Mitt Romney, Briana Mihok is gold dust. She ticks the three boxes that qualify her to be exactly the type of voter the presidential candidates are focusing all their attention on as the White House race enters its final three weeks. She is undecided, she lives in an unexpectedly competitive state and – by far the most important of all – Mihok is a woman.

Presidential elections always come down to a narrow territorial battle for advantage in a handful of swing states. But after almost a year of constant campaigning and millions of dollars of television advertising, it became clear this week that both candidates believe that the defining battle of 2012 is now for female voters.

At the second presidential debate on Tuesday night on Long Island, Obama and Romney clashed repeatedly on women's issues, ranging from workplace equality to access to contraception. Romney said women had suffered worse under Obama's economic policies, while the president in turn tied issues like equal pay and healthcare to economic recovery, describing women as important breadwinners in their family.

A look at the polls over the last two weeks explains why. Since the first debate, Romney, whose campaign had been faltering, has surged back into contention, overtaking Obama in some national polls. Behind those headline figures is the first sign of a big shift among women, with Obama's massive advantage cut back to a few points.

But for some of the women whose votes the campaigns so openly covet, the sudden intensity of these arguments has done little to clarify matters.

Mihok, a mother of one and registered Republican, said she was struck by Romney's ignorance of issues that concern women during Tuesday's debate.

She cited the answer he gave to a question about workplace equality in which he said he understood the need to be "flexible" in order to have women in the workforce. Giving the example of his chief of staff, who had two children of school age, Romney told the audience at Hofstra university: "She said: 'I can't be here until 7 or 8 o'clock at night: I need to be able to get home at 5 o'clock so I can be there for making dinner for my kids and being with them when they get home from school.' So we said: 'Fine. Let's have a flexible schedule so you can have hours that work for you.'"

Romney's comments were "weird", said Mihok, a 32-year-old policy analyst for a non-profit research company at the University of Pittsburgh. She has been leaning towards Obama in recent weeks.

"My husband and I both work, and flexible work hours are just as important to my husband as they are to me," she said. "Unfortunately governor Romney showed a real ignorance when it comes to women's issues."

Mihok, who did not watch Tuesday's debate but discussed it with friends and read about it afterwards, said Obama put forward a better case for women. "President Obama, with his background and his upbringing, has the opportunity to see first hand what is important to women and their families. Look at the statement about his grandmother training men for positions for which men got higher pay and better promoted, yet she never had that opportunity. That legacy is still around, and it's important to recognise that."

But while that's good news for Obama, Mihok said women's issues would not be at the front of her mind when she came to cast her ballot on 6 November.

She is not alone. Mihok is one of a number of undecided female voters from Pennsylvania whom the Guardian is talking to over the course of the election to find out what is important to them. None of them cited equal pay or social issues as top of the list of things important to their vote, although three out of five cited affordable healthcare as ranking top alongside the economy.

"Who can say what women need or want?" said Judy Beck, 52, a grandmother and divorced mother of three from O'Hara township in Allegheny County. "We are really just people."

The issues most important to Beck's vote are the economy, foreign policy and the partisan makeup of Congress, but she said she did not hear anything on any of these issues that would make her change her mind, she said.

Shireen Parsons, 68, a grandmother and community organiser from Summit Hill, Carbon County, was also unconvinced by the efforts to appeal to women. "I'm old and cranky and sceptical," said Parsons, an independent. "Women voters are smarter than most politicians think. When Romney talked about women, just how many women did he have in his cabinet? I've no idea but I would like to know."

'What decade does Mitt Romney think he's living in?'

Pennsylvania is typically Democratic but the race for its 20 electoral college votes has tightened considerably since the first debate.

It has a particularly marked gender divide among voters, as well as a higher pay disparity among men and women. Women earn almost 81¢ to a male worker's dollar, compared to the national average of 82¢.

Of the five women, three are Republican, one is a Democrat and one an independent. In the last two weeks, one of the Republicans has decided to vote for Obama, with another two leaning that way. But this week, the Democrat said she is leaning heavily towards Romney.

Beck, who is "principally a Republican" because of her belief in less government, has made up her mind to vote for Obama over the course of the campaign.

She said that the most recent arguments over women from the candidates has "solidified my leaning towards the Obama campaign".

"Obama didn't get as specific as Mitt Romney. Romney was too specific and it was apparent he doesn't have a clue. Obama mentioned his mother was a single mother, and he understands there are single mothers who need to make a living and that both men and women need to work."

Obama answered the equal pay question during Tuesday's debate by referring to his upbringing by a single mother and grandmother who trained men for jobs that paid them more and ranked them higher. He cited the Lilly Ledbetter Fair Pay Act, which makes it easier for women to sue over pay discrimination, as one of his first official acts in office. And he cited childcare tax credits as a factor in whether women can become the breadwinners.

"These are not just women's issues," the president said. "These are family issues. These are economic issues. And one of the things that makes us grow as an economy is when everybody participates and women are getting the same fair deal as men are."

These comments chimed with the Pennsylvania women. Romney's answer to the same question was to insist that his record when governor or Massachusetts was one of equality. His description of his efforts to hire women into his cabinet were pilloried as wrong-footed and false, and he has been widely ridiculed for his "binders full of women" remark about female appointments to his cabinet.

Beck said that Romney reminds her of an old boss who refused her a raise because he told her "the married men need it first".

"That was back in 1987," she said. "It was wrong then, and it is so much more wrong now. Any suggestion that a woman should be paid any differently is ridiculous. What decade does Mitt Romney think he's living in?

"Romney does not like anyone to disagree with him, You need to be the president but you also need to listen to advice. It seems to me he assumes he has all the answers."

Like Beck, Parsons said she found herself being more sceptical of Romney than Obama.

"Romney believes that life begins at conception, but he said he wouldn't legislate against a woman's right to abortion. I don't believe that," she said.

"Because of Obama's upbringing and who he is and also because of his wife, he tends to do better among women. I'm inclined to trust some things that Obama says than anything Mitt Romney says. I'm not sure why.

"On the other hand I see what Obama has failed to do for women at the bottom layer. He is as much owned by corporate interests as other politicians. They work for corporate interests and leave poor people women and children at the bottom of the heap."

Joanne Quinn-Smith, 63, a talkshow host, publisher of an online news network and CEO of a marketing company, praised Obama for talking about women as breadwinners. "If I work as hard as a man I want to get paid like a man," she said.

She also pointed out the gender pay disparity in Pennsylvania, but said the economy was her number one concern.

"I want someone who can bring us out of this (recession)," she said. "Someone who wants to drive business or someone who wants to drive spending that is supported by taxes? I think I just told you where I am leaning. Someone needs to say: 'We've spent enough money, let's re-organise, let's tighten our belts.' Who is better to do that than someone from a business environment?"

Quinn-Smith said that she thought that Romney would be successful in gaining votes among women in higher income brackets.

But, despite leaning towards the Republican, she is still not sure. "In the last Guardian piece someone said, it is about who you hate less."

She compared the election to a college student "going home for the weekend and talking to your parents for advice, and you need them to say something wonderful about what's going on and you don't feel any better because you haven't gotten it."